ES+(2,2):D5 - GroupNames

G = 2₊¹⁺⁴⋊D₅ order 320 = 2⁶·5

1^st semidirect product of 2₊¹⁺⁴ and D₅ acting via D₅/C₅=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: 2₊¹⁺⁴⋊₁D₅, (C₅×D₄)⋊₁₅D₄, (C₅×Q₈)⋊₁₅D₄, C₂₀⋊D₄⋊₈C₂, C₅⋊₅(D₄⋊₄D₄), D₄⋊₇(C₅⋊D₄), C₄○D₄.₉D₁₀, Q₈⋊₇(C₅⋊D₄), D₄⋊D₁₀⋊₅C₂, (C₂×D₄).₈₂D₁₀, C₂₀.₂₁₇(C₂×D₄), C₁₀.₈₀C₂²≀C₂, C₂₀.D₄⋊₁₁C₂, (C₂×D₂₀)⋊₁₅C₂², (C₂×C₂₀).₂₁C₂³, (C₄×Dic₅)⋊₈C₂², (C₂²×C₁₀).₂₄D₄, D₄⋊₂Dic₅⋊₁₁C₂, C₂³.₁₂(C₅⋊D₄), C₄.Dic₅⋊₁₀C₂², (C₅×2₊¹⁺⁴)⋊₁C₂, (D₄×C₁₀).₁₀₇C₂², C₂.₁₄(C₂⁴⋊₂D₅), C₄.₆₄(C₂×C₅⋊D₄), (C₂×C₁₀).₄₂(C₂×D₄), (C₂×C₄).₂₁(C₂²×D₅), C₂².₁₄(C₂×C₅⋊D₄), (C₅×C₄○D₄).₁₉C₂², SmallGroup(320,868)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₀ — 2₊¹⁺⁴⋊D₅

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂×C₂₀ — C₂×D₂₀ — D₄⋊D₁₀ — 2₊¹⁺⁴⋊D₅

Lower central

C₅ — C₁₀ — C₂×C₂₀ — 2₊¹⁺⁴⋊D₅

Upper central

C₁ — C₂ — C₂×C₄ — 2₊¹⁺⁴

Generators and relations for 2₊¹⁺⁴⋊D₅
G = < a,b,c,d,e,f | a⁴=b²=d²=e⁵=f²=1, c²=a², bab=faf=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf=ab, dcd=fcf=a²c, ce=ec, de=ed, fdf=cd, fef=e^-1 >

Subgroups: 670 in 168 conjugacy classes, 43 normal (17 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, C₂³, C₂³, D₅, C₁₀, C₁₀, C₄², M₄(2), D₈, SD₁₆, C₂×D₄, C₂×D₄, C₄○D₄, C₄○D₄, Dic₅, C₂₀, C₂₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₄.D₄, C₄≀C₂, C₄⋊₁D₄, C₈⋊C₂², 2₊¹⁺⁴, C₅⋊₂C₈, D₂₀, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₅×D₄, C₅×Q₈, C₂²×D₅, C₂²×C₁₀, C₂²×C₁₀, D₄⋊₄D₄, C₄.Dic₅, C₄×Dic₅, D₄⋊D₅, Q₈⋊D₅, C₂×D₂₀, C₂×C₅⋊D₄, D₄×C₁₀, D₄×C₁₀, C₅×C₄○D₄, C₅×C₄○D₄, C₂₀.D₄, D₄⋊₂Dic₅, C₂₀⋊D₄, D₄⋊D₁₀, C₅×2₊¹⁺⁴, 2₊¹⁺⁴⋊D₅
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, C₂×D₄, D₁₀, C₂²≀C₂, C₅⋊D₄, C₂²×D₅, D₄⋊₄D₄, C₂×C₅⋊D₄, C₂⁴⋊₂D₅, 2₊¹⁺⁴⋊D₅

Smallest permutation representation of 2₊¹⁺⁴⋊D₅
►On 40 points

Generators in S₄₀

(1 14 9 19)(2 15 10 20)(3 11 6 16)(4 12 7 17)(5 13 8 18)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)

(1 39)(2 40)(3 36)(4 37)(5 38)(6 31)(7 32)(8 33)(9 34)(10 35)(11 26)(12 27)(13 28)(14 29)(15 30)(16 21)(17 22)(18 23)(19 24)(20 25)

(1 19 9 14)(2 20 10 15)(3 16 6 11)(4 17 7 12)(5 18 8 13)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)

(1 39)(2 40)(3 36)(4 37)(5 38)(6 31)(7 32)(8 33)(9 34)(10 35)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)

(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)(31 32 33 34 35)(36 37 38 39 40)

(1 8)(2 7)(3 6)(4 10)(5 9)(12 15)(13 14)(17 20)(18 19)(21 36)(22 40)(23 39)(24 38)(25 37)(26 31)(27 35)(28 34)(29 33)(30 32)

G:=sub<Sym(40)| (1,14,9,19)(2,15,10,20)(3,11,6,16)(4,12,7,17)(5,13,8,18)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (1,39)(2,40)(3,36)(4,37)(5,38)(6,31)(7,32)(8,33)(9,34)(10,35)(11,26)(12,27)(13,28)(14,29)(15,30)(16,21)(17,22)(18,23)(19,24)(20,25), (1,19,9,14)(2,20,10,15)(3,16,6,11)(4,17,7,12)(5,18,8,13)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (1,39)(2,40)(3,36)(4,37)(5,38)(6,31)(7,32)(8,33)(9,34)(10,35)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40), (1,8)(2,7)(3,6)(4,10)(5,9)(12,15)(13,14)(17,20)(18,19)(21,36)(22,40)(23,39)(24,38)(25,37)(26,31)(27,35)(28,34)(29,33)(30,32)>;

G:=Group( (1,14,9,19)(2,15,10,20)(3,11,6,16)(4,12,7,17)(5,13,8,18)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (1,39)(2,40)(3,36)(4,37)(5,38)(6,31)(7,32)(8,33)(9,34)(10,35)(11,26)(12,27)(13,28)(14,29)(15,30)(16,21)(17,22)(18,23)(19,24)(20,25), (1,19,9,14)(2,20,10,15)(3,16,6,11)(4,17,7,12)(5,18,8,13)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (1,39)(2,40)(3,36)(4,37)(5,38)(6,31)(7,32)(8,33)(9,34)(10,35)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40), (1,8)(2,7)(3,6)(4,10)(5,9)(12,15)(13,14)(17,20)(18,19)(21,36)(22,40)(23,39)(24,38)(25,37)(26,31)(27,35)(28,34)(29,33)(30,32) );

G=PermutationGroup([[(1,14,9,19),(2,15,10,20),(3,11,6,16),(4,12,7,17),(5,13,8,18),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40)], [(1,39),(2,40),(3,36),(4,37),(5,38),(6,31),(7,32),(8,33),(9,34),(10,35),(11,26),(12,27),(13,28),(14,29),(15,30),(16,21),(17,22),(18,23),(19,24),(20,25)], [(1,19,9,14),(2,20,10,15),(3,16,6,11),(4,17,7,12),(5,18,8,13),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40)], [(1,39),(2,40),(3,36),(4,37),(5,38),(6,31),(7,32),(8,33),(9,34),(10,35),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30),(31,32,33,34,35),(36,37,38,39,40)], [(1,8),(2,7),(3,6),(4,10),(5,9),(12,15),(13,14),(17,20),(18,19),(21,36),(22,40),(23,39),(24,38),(25,37),(26,31),(27,35),(28,34),(29,33),(30,32)]])

50 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	5A	5B	8A	8B	10A	10B	10C	···	10T	20A	···	20L
order	1	2	2	2	2	2	2	2	4	4	4	4	4	4	5	5	8	8	10	10	10	···	10	20	···	20
size	1	1	2	4	4	4	4	40	2	2	4	4	20	20	2	2	40	40	2	2	4	···	4	4	···	4

50 irreducible representations

dim

type

image

C₁

C₂

D₄

D₅

D₁₀

C₅⋊D₄

D₄⋊₄D₄

2₊¹⁺⁴⋊D₅

kernel

2₊¹⁺⁴⋊D₅

C₂₀.D₄

D₄⋊₂Dic₅

C₂₀⋊D₄

D₄⋊D₁₀

C₅×2₊¹⁺⁴

C₅×D₄

C₅×Q₈

C₂²×C₁₀

2₊¹⁺⁴

C₂×D₄

C₄○D₄

D₄

Q₈

C₂³

C₅

C₁

# reps

Matrix representation of 2₊¹⁺⁴⋊D₅ ►in GL₆(𝔽₄₁)

40	0	0	0	0	0
0	40	0	0	0	0
0	0	0	40	0	0
0	0	1	0	0	0
0	0	0	0	0	40
0	0	0	0	1	0

40	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	0	40
0	0	0	0	40	0
0	0	0	40	0	0
0	0	40	0	0	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	0
0	0	40	0	0	0
0	0	0	0	0	40
0	0	0	0	1	0

40	0	0	0	0	0
0	40	0	0	0	0
0	0	0	0	0	40
0	0	0	0	1	0
0	0	0	1	0	0
0	0	40	0	0	0

10	0	0	0	0	0
0	37	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	18	0	0	0	0
16	0	0	0	0	0
0	0	1	0	0	0
0	0	0	40	0	0
0	0	0	0	0	1
0	0	0	0	1	0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1,0,0,0,0,40,0,0,0],[10,0,0,0,0,0,0,37,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,16,0,0,0,0,18,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

2₊¹⁺⁴⋊D₅ in GAP, Magma, Sage, TeX

2_+^{1+4}\rtimes D_5

% in TeX

G:=Group("ES+(2,2):D5");

// GroupNames label

G:=SmallGroup(320,868);

// by ID

G=gap.SmallGroup(320,868);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,570,1684,851,438,102,12550]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^4=b^2=d^2=e^5=f^2=1,c^2=a^2,b*a*b=f*a*f=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=a*b,d*c*d=f*c*f=a^2*c,c*e=e*c,d*e=e*d,f*d*f=c*d,f*e*f=e^-1>;

// generators/relations

G = 2+ 1+4⋊D5 order 320 = 26·5

1st semidirect product of 2+ 1+4 and D5 acting via D5/C5=C2

G = 2₊¹⁺⁴⋊D₅ order 320 = 2⁶·5

1^st semidirect product of 2₊¹⁺⁴ and D₅ acting via D₅/C₅=C₂